gauss flows

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AIM

To understand in particular, the mathematical formulation of load flow model in complex form

and a simple method of solving load flow problems of small sized system using Gauss-Seidel iterative

algorithm.

OBJECTIVES

i. To write a computer program to solve the set of non-linear load flow equations using Gauss-Seidel

Load Flow (GSLF) algorithm and present the results in the format required for system studies.

ii. To investigate the convergence characteristics of GSLF algorithm for normally loaded small

system for different acceleration factors.

iii. To investigate the effects on the load flow results, load bus voltages and line / transformer

loading, due to the following control actions:

a. Variation of Voltage settings of P-V buses

b. Variation of shunt compensation at P-Q buses

c. Variation of tap settings of transformer

d. Generation shifting or rescheduling

SOFTWARE REQUIRED

ETAP.

THEORY

Need For Load Flow Analysis

Load Flow analysis, is the most frequently performed system study by electric utilities. This

analysis is performed on a symmetrical steady-state operating condition of a power system under

normal mode of operation and aims at obtaining bus voltages and line / transformer flows for a given

load condition. This information is essential both for long term planning and next day operational

planning. In long term planning, load flow analysis, help in investigating the effectiveness of alternative

plans and choosing the best plan for system expansion to meet the projected operating state. In

operational planning, it helps in choosing the best unit commitment plan and generation schedules to run

the system efficiently for the next days load condition without violating the bus voltage and line flow

operating limits.

LOAD FLOW ANALYSIS-GAUSS SEIDAL METHODEx.No :05

Date :02.09.10


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Description of Load Flow Problem

In the load flow analysis, the system is considered to be operating under steady state balanced

condition and per phase analysis is used. With reasonable assumptions and approximations, a power

system under this condition may be represented by a power network as single-line diagram.

The Network consists of a number of buses (nodes) representing either generating stations or bulk

power substations, switching stations interconnected by means of transmission lines or power

transformers. The bus generation and demand are characterized by complex power flowing into and out

of the buses respectively. Each transmission line is characterized by its equivalent circuit. The

transformer with off-nominal tap ratio is characterized by their equivalent circuit. Shunt compensating

capacitors or reactors are represented as shunt susceptance.

Load flow analysis is essentially concerned with the determination of complex bus voltages at all

buses, given the network configuration and the bus demands. Let the given system demand (sum of all

bus demands) be met by a specific generation schedule. A generation schedule is nothing but a

combination of MW generation (chosen within their ratings) of the various spinning generators the total

of which should match the given system demand plus the transmission losses. It should be noted that

there are many generation schedules available to match the given system demand and one such schedule

is chosen for load flow analysis.

The Ideal Load flow problem is stated as follows:

Given: The network configuration (Bus admittance matrix) and all the bus power injections (bus injection

refers to bus generation minus bus demand).

To determine: The complex voltages at all the buses.

The steady state of the system is given by the state vector X defined as

Once the state of the system is known, all the other quantities of interest in the power network can be

computed.


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Development of Load Flow Model

The Load flow model in complex form is obtained by writing one complex power matching

equation at each bus.

Fig 1 Complex Power Balancing at a Bus

Referring to Fig 1 the complex power injection (generation minus demand) at the kth bus is equal to the

complex power flowing into the network at that bus which is given by

PIk+ jQIk= Pk+ jQk (1)

In expanded form

(PGk- PDk) + j (QGk- QDk) = VkIk* (2)

The network equation relating bus voltage vector V with Bus current vector I is

YV = I (3)

Taking the kth

component of I from (3) and substituting for Ik*

in (2) we get the power flow model incomplex form as

K = 1,2,.N (4)

In (4) there are N complex variable equations from which the N unknown complex variables V1,.VN

can be determined.

Transmission line/ Transformer Flow equation

In a Load Flow package after solving equation (4) for complex bus voltages using any iterative

method, the active and reactive power flows in all the lines/ transformers are to be computed. A common

n equivalent circuit for transmission line and transformer is given in Fig 3.2. For a transmission line set

the variable "a" equal to unity and for a transformer set variable be equal to zero. The expression for

power flow in line / transformer k-m from the kth bus to the mth bus, measured at the kth bus end is given

by (refer Fig 2)


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Fig 2 PI Equivalent Circuit of a Transmission Line / Transformer

** ttkkkmkm IVIVjQP ==+ (5)

Noting that

aVV tk =/ (6)

)()( ctkmmtt jbVYVVI += (7)

Substituting equations (3.6) and (3.7) in equation (3.5) we get

*)()/(**)])/*)[/( 2 ckkmmkkkmkm jbaVYVaVaVjQP +=+ (8)

Similarly the power flow in line k-m from the mth bus to kth bus measured at the mth bus end is.

*)(*)]/*(*[* 2 cmkmkmmmmmkmk ibVYaVVVIVJQP +==+ (9)

The complex power loss in line/transformer k-m, PLkm+jQLkm, is given by the sum of the two expressions

(8) and (9)

Classification of Buses

From the Load Flow model in equation (4) and from the definition of complex bus the two

expressions (8) and (9)

Classification of Buses

From the Load Flow model in equation (4) and from the definition of complex bus voltage, Vk as

kkk LVV =

One can observe that there are four variables, PI, QI, and V associated with each bus. Any two of

these four may be treated as independent variables (that is specified) while the other two may be

computed by solving power flow equations. The buses are classified based on the variables specified.

Three types of buses classified based on practical requirements are given below :

Slack Bus : While specifying a generation schedule for a given system demand, one can fix up the

generation setting of all the generation buses except one bus because of the limitation of not knowing the

transmission loss in advance. This leaves us with the only alternative of specifying two variables s and

Vs pertaining to a generator bus (usually a large capacity generation bus is chosen and this is called as


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slack bus) and solving for the remaining (N-1) complex bus voltages from the respective (N-1) complex

load flow equations. Incidentally the specification of Vs helps us to fix the voltage level of the system

and the specification ofs as zero, makes Vs as reference phasor. Thus for the slack bus, both and V

are specified and PG and QG are to be computed only after the iterative solution of bus voltages is

completed.

P-V buses : In order to maintain a good voltage profile over the system, it is customary to maintain the

bus voltage magnitude of each of the generator buses at a desired level. This can be achieved in practice

by proper Automatic Voltage Regulator (AVR) settings. These generator buses and other Voltage-

controlled buses with controllable reactive power source such as SVC buses are classified as P-V buses

since PG and V are specified at these buses. Only one state variable, is to be computed at this bus. The

reactive power generation QG at this bus which is a dependent variable is also to be computed to check

whether it lies within its operating limits.

P-Q buses : All other buses where both PI and QI are specified are termed as P-Q buses and the these

buses both and V are to be computed.

Hence the Practical Load Flow problem may be stated as :

Given : The network configuration (bus admittance matrix), all the complex bus power demands, MW

generation scheduled and voltage magnitudes of all the P-V buses, and voltage magnitude of the slack

bus,

To determine : The bus voltage phase angles of all buses except the slack bus and bus voltage

magnitudes of all the P-Q buses.

Hence the state vector to be solved from the Load Flow model isT

NQNP VVVX )................( .2121 =

where NP = N-1

NQ=N-NV-1

and the NV number of P-V buses and the slack bus are arranged at the end.

Solution to Load Flow Problem

A number of methods are available for solving Load Flow problem. In all these methods, voltage

solution is initially assumed and then improved upon using some iterative process until convergence is

reached. The following three methods will be presented :

(i) Gauss-Seidel Load Flow (GSLF) method


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(ii) Newton-Raphson Load Flow (NRLF) method

(iii) Fast Decoupled Load Flow (FDLF) method

The first method GSLF is a simple method to program but the voltage solution is updated only node

by node and hence the convergence rate is poor. The NRLF and FDLF methods update the voltage

solution of all the buses simultaneously in each iteration and hence have faster convergence rate.

Taking the complex conjugate of equation (4) and transferring Vk to the left hand side, we obtain

;/]1

*/)[( kkmkm YVYm

NVkjQIkPIkVk

==

k

k = 1, 2, (N-1) (Slack bus excluded) (10)

Define kkkk YjQIPIAk /)( = (11)

kkkmkm YYB /= (12)

The voltage equation to be solved during the h (h) iteration of G.S method is obtained from (10), (11) and

(12) as

)()1()()1(

11

1*)/(

h

mkm

h

mkm

h

kk

h

k VBkm

NVB

m

kVAV

+=

=

=

++(13)

GSLF Algorithm

The algorithm for GSLF is given in the flow chart.

Convergence Check :

Referring to Flow chart , during every iteration h, the maximum change in bus voltage that has

occurred is stored in VMAX as given below

VMAX = max sNkfeh

k

h

k =++

;.......2,1;,)1()1(

(14)

Where

)()1()1()1()1( h

k

h

k

h

k

h

k

H

k VVfjeV =+

++=+

The convergence is checked by comparing VMAX with the specified tolerance .

Additional Computation for P V Bus

The flow chart does not have provision for voltage controlled buses. However, if the link

between X and Y in Flow chart is removed and the P-V bus module in next flowchart is introduced,

then P V buses can be handled.

Referring to Fig 3 and Fig.4, for each P-V bus during the h th iteration, before updating bus voltage,

the following computations are made :

Step 1:


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Adjusting the complex voltage)()()( h

k

h

k

h

k jfeV + to correct the voltage magnitude to the scheduled

value,schk

V as follows

)/tan()()()( h

k

h

k

h

k efare= (15)

)()(

)(

hk

schk

h

newkjeVV = (16)

Step : 2

Compute the reactive power generation using the Vk(new)(h) as

)( )()()( hh

kk

h

k VQQDQG += (17)

++=

= +=

+1

1 1

)()()(

)1()(

)(

)()( *Im)(k

m

N

km

h

mkm

hnew

k

kk

h

mkm

h

newk

hh

k VYVYVYVagVQ

If the inequality max)( khkmmk QGQGQG is satisfied, the Vk(h) is set as V(h)k(new).

Go to step 3.

If QGk(h)>QGk

max, then set QGk(h) = QGk

max, go to step 3.

If QGk(h)


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Start

Read data

Form Y Bus

Compute Ak

and Bkm

;k = 1, 2 .. N; s

m = 1, 2, N using equations 11 & 12

Set iteration count

h = 0

Set Bus count

k = 1

Update voltage Vk

(h+1)

using equation (13)

k=k+1

X

Y

Compute line flows, line loss, slack bus

power and print result

Stop

Checkconvergence

IsVMAX


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yes

IS QGk

(h) < QGk

(min)

A P-

Vbus

Compute V(h)knew

using(16)

Compute QGk

(h) using(17)

Is QGk

(h) > QGk

(max)

Vk

(h) =V(h)knew

QGk

(h) < QGk

(min)

QGk

(h) > QGk

(max)

Recompute Akusing(11) and QG

k

(h)

y

Yes No

No


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NEWTON-RAPHSON AND FAST-DECOUPLED METHODS:

LOAD FLOW MODEL IN REAL VARIABLE FORM

Referring to fig 1, the complex power balance at bus k is given by

(PIk+ jQIk) = Pk + jQk ..(1)

where the complex power injection at the kth bus (PIk + jQIk) is equal to the complex power flowing into

the network through all the lines connected to kth bus (PIk + jQIk). Since the bus generation and demand

are specified, the complex power injection is a specified quantity and is given by

PIk(sp) + jQIk(sp) = (PGk(sp)-PDk(sp)) + j(QGk(sp)-QDk(sp)) ..(2)

The complex power (Pk + jQk) can be written as a function of state variables as

Pk + jQk= VkIk* = VkY*km V*m ..(3)

Substituting Vk= |Vk| kand Ykm = |Ykm| km, equation (3) becomes

Pk(,V) = |Vk| |Ykm| |Vm| cos (k - m - km) ..(4)

Qk(

,V) = |Vk|

|Ykm| |Vm| sin (

k-

m -

km) ..(5)

Substituting equations (2), (4), (5) in equation (1), we get the following two power balance equations for

kth bus.

Pk(,V) - PIk(sp) = 0 ..(6)

Qk(,V) QIk(sp) = 0 ..(7)

Load Flow model in real variable form is compiled by adopting the following rules

(i) For every bus phase angle is unknown, include the respective real power balance equation

(6)

(ii) For every bus whose bus voltage magnitude |V| is unknown, include the respective reactive

power balamce equation (7).

Thus for a system with a slack bus and M P-V buses, the number of rreal power balance equation, NP is

given by

NP = N-1

And the number of reactive power balance equations, NQ is given by

NQ = N-M-1

Assuming that the buses are numbered with P-Q buses first, followed by P-V buses and then by the slack

bus, the Load Flow model for the system with M P-V buses is given by

(PGk+jQG

k)

G(PD

k+jQD

k)

(PIk+jQI

k) = (PG

k-PD

k)

+ j(QGk-QD

k)

k Vk

k Vk

(Pk+jQ

k) (P

k+jQ

k)I

kI

k

Fig 1(a) Fig 1(b)


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fk= Pk (,V) - PIk(sp) = 0; k = 1,2,..NP ..(8)

fk+NP = Qk (,V) QIk(sp) = 0; k = 1,2,..NQ ...(9)

where expressions Pk (,V) and Qk (,V) are given by equations (4) and (5). Equations (8) and (9) can be

written in compact form as

f(X) = 0 ..(10)

Where X = state vector = (1, 2,..NP,V1, V2,..VNQ)T

and the number of nonlinear equations in F in equation (10) is equal to (NP+NQ)


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2.Computer mismatch vector using equation (15) and (16)

3. |Pi| ; i = 1,2,..NP

Compute PQMAX max

|Qj| ; j = 1,2,...NQ

If PQMAX (the tolerance) stop. Otherwise go to step 4.

4. Compute Jacobian matrix using equation (18), (19), (20) and (21).

5. Obtain state correction vector by solving equation (13) using optimally ordered

triangular factorization.

6. Update state vector using equation (17). Go to step 2.

FAST-DECOUPLED LOAD FLOW METHOD

Fast-Decoupled Load Flow (FDLF) method is faster, simpler to program, equally reliable and

requires less memory than NRLF method. The model for this method is developed from NRLF method by

employing the P- / Q-V decoupling principle.

The starting point for the derivation of the FDLF model is the voltage correction scheme of the Newton-

Raphson method given in equation (13).

The first step in applying the P- / Q-V decoupling principle is to neglect the coupling submatrices [N]

and [J] in equation (13) which results in two separate equations

[H] = P ..(22)

[L] |V| / |V| = Q ..(23)

The second step is to make certain physically justifiable simplifications. In practical power systems, the

following assumptions are almost always valid.

coskm = 1Gkm sinkm


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|V2| (-B22) |V2| |V2| (-B23) |V3| |V2| (-B24) |V4| |V2| (-B25) |V5| 2 P2|V3| (-B32) |V2| |V3| (-B33) |V3| |V3| (-B34) |V4| |V3| (-B35) |V5| 3 P3

=

|V4| (-B42) |V2| |V4| (-B43) |V3| |V4| (-B44) |V4| |V4| (-B45) |V5| 4 P4

|V5| (-B52) |V2| |V5| (-B53) |V3| |V5| (-B54) |V4| |V5| (-B55) |V5| 5 P5

..(26)

|V2| (-B22) |V2| |V2| (-B23) |V3| |V2| (-B25) |V5| |V2| / |V2| Q2|V3| (-B32) |V2| |V3| (-B33) |V3| 0 |V3| / |V3| = Q3

|V5| (-B52) |V2| 0 |V5| (-B55) |V5| |V5| / |V5| Q5

..(27)

The decoupling process is completed by the following approximations

(a) Omitting from [B] the representation of those network elements that predominantly affect MVAR

flows, i.e shunt reactances and off-nominal in-phase transformer taps (tap ratio is taken as 1 p.u.)

(b) Omitting from [B] the angle shifting effects of the phase shifters (phase shifter ratio taken as 1

p.u.)

(c) In (26) and (27), the voltage magnitude terms appearing on the left side of each element of matrix

are transferred to the right hand side of the equation.

(d) The voltage magnitude term appearing on the right of each element of the matrix in (26) are set as1 p.u.

(e) In equation (27), the voltage magnitude terms appearing on the right side of each element are

eliminated by cancelling the voltage magnitude terms appearing in the incremental vector|V| / |

1 2 3

45

Slack bus P-Q bus

P-Q bus

p

P-V bus

P-Q bus

pFig 2

Five bus sample system


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V|

(f) In equation (26), the susceptance elements are calculated neglecting the resistances.

With the above approximations, the final fdpf model becomes

[B] = P / |V|

[B] |V| = Q / |V|

Both [B] and [B] are real, sparse matrices and have the structures of [H] and [L]

respectively.

FDLF Algorithm

Equations (28) and (29) are solved alternatively, always using the most recent voltage values. Each

iteration cycle comprises one solution for to update and then one solution for|V| to update |V|.

Separate convergence tests are used for (28) and (29) as

PMAX max {|Pi|; i = 1,2,..NP} pPMAX max {|Qj|; j = 1,2,..NQ} P

ADDITIONAL COMPUTATION FOR P-V BUS

For each P-V bus, before computing [Q / |V|] fig 3, during each iteration, the reactive power generation

at tha bus, QGk, is computed using eqn (30) and if it violates either the upper or lower limit, then the

scheduled voltage magnitude is corrected using sensitivity factor to contain the QG kwithin operating

limits. The sensitivity factors are computed and stored before the start of iteration process. The flow chart

for P-V module is given in fig 4.


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Is QGk

(h)QGk

(max)

Fig 4 . P-V module for FDLF Algorithm


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FLOWCHART FOR FDLF ALGORITHM:

yes

Assume flat start for and V.Compute [B] & [B] and factorise

KP = KQ = 1

Compute [P / |V|]

KQ = 1

COMPUTE [Q / |V|]

SOLVE (25) AND UPDATE V

KP = 1

Is

KQ=0

OUTPUT

Is

KP=0

Fig 3

yes

yes

yes

Is PMAX p

KP = 0

Is QMAX q

KQ = 0

no

no

no

no

SOLVE (24) AND UPDATE


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Question

1.Five bus system. Generators are connected at buses 1 and 3. loads are indicated at buses2,4,and 5. base

values for the system are 100MVA,138kV in the high tension lines considered here. Table 1 gives

impedance for the lines, which are identified by the buses on which they terminate. The charging

megavars listed in the table account for the distributed capacitance of the lines.

Write a program for load flow analysis using gauss seidal method.

TABLE 1

TABLE 2

Bus Generation Load V

P MW Q MW P MW Q MW PU

1 --- --- 65 30 1.04 0

2 0 0 115 60 1.00 0

3 180 --- 70 40 1.02 0

4 0 0 70 30 1.00 0

5 0 0 85 40 1.00 0

2.Create necessary changes in above program for different values of acceleration factor. Determine

Best acceleration factor.

3.Create necessary changes in above program for different values of Voltage specified for generator

G1 (1.0 to 1.04 PU) and comment on the voltage magnitude of the load buses and transmission system

losses.

4.Create necessary changes in above program shifting bulk generation from slac bus 1 to bus 2.

Line Length R X Charging MVar

SB EB In Km Ohms Ohms At 138kV

1 2 64.4 8 32 4.1

1 5 48.3 6 24 3.1

2 3 48.3 6 24 3.1

3 4 128.7 16 64 8.2

3 5 80.5 10 40 5.1

4 5 96.5 12 48 6.1

gauss flows

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